Multi-Temperature Mixture of Fluids

We present a survey on recent results concerning some different models of a mixture of compressible fluids. In particular, we discuss the most realistic case of a mixture where each constituent has its own temperature (MT). We first compare the solutions of this model with the one with unique common temperature (ST). In the case of Eulerian fluids, it will be shown that the corresponding ST differential system is a principal subsystem of the MT system. Global behavior of smooth solutions for large time for both systems will also be discussed through the application of the Shizuta-Kawashima K-condition. Then we introduce the concept of the average temperature of a mixture based on the consideration that the internal energy of the mixture is the same as that in the case of a single-temperature mixture. As a consequence, it is shown that the entropy of the mixture reaches a local maximum in equilibrium. Through the procedure of the Maxwellian iteration, a new constitutive equation for nonequilibrium temperatures of constituents is obtained in a classical limit, together with the Fick law for the diffusion flux. Finally, in order to justify the Maxwellian iteration, we present, for dissipative fluids, a possible approach to a classical theory of mixtures with the multi-temperature. We prove that the differences of temperatures between the constituents imply the existence of a new dynamic pressure even if fluids have zero bulk viscosities.